Connectedness is a fundamental property of objects and systems. It is usually viewed as inherently topological, and hence treated as derived property of sets in (generalized) topological spaces. There have been several independent attempts, however, to axiomatize connectedness either directly or in the context of axiom systems describing separation. In this review-like contribution we attempt to link these theories together. We find that despite differences in formalism and language they are largely equivalent. Taken together the available literature provides a coherent mathematical framework that is not only interesting in its own right but may also be of use in several areas of computer science from image analysis to combinatorial optimization.